I'm reframing an existing stairway opening to allow for additional head height at the bottom of the stairs. I would have to cut through a joist to extend the opening an extra 16" and create a new double header along the side with new lumber. The existing double header along the run of the stairs is 2 pieces of old lumber with actual dimensions of 2.5" x 8-3/8" and 7 ft long which gives an actual width of 5" since they are sistered. Obviously New lumber doesn't come in these dimensions so the closest I can get is using 4 2x10's with a new span of 8'4". I would assume old lumber is stronger than 2x's so is there some kind of conversion chart or is it really as simple as matching dimensions? I'm only extending another 16" so I'm not adding much as far as load goes.

I think for extending your current span by 16", adding the additional thickness at the same depth should be fine. Unless there is visible sagging now , I wouldn't be concerned.
If concerned with the span, LVL headers could be used. There are plenty of span charts available. Often lumber yards will help you to calculate simple openings.